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What Is the Circle Scribe Disk Compass?

Stage: 3 and 4

Article by Bill Harper

Published May 2001,February 2011.

It is a new geometrical instrument and a toy with 3 basic capabilities:

It draws circles and arcs of circles with unparalleled ease and accuracy

It accurately constructs and measures angles in a way which demonstrates the rotational nature of angle

It plots any point under the disk to within 0.5mm using a polar co-ordinate system

The disk spins around a steel pin through its centre and has 100 holes arranged along a spiral. Each one is 1 mm further from the centre than the last. The protractor scale around the edge measures its rotation from alignment with one line to alignment with another. Circle Scribe claims that the disk compass is the most significant step in geometrical instrumentation since Thales of Miletus
returned to Greece after his studies with the Pyramid builders of Egypt 500 BC.

You can make your own by printing out a spiral onto an overhead transparency. Use Excel or some other spreadsheet to create the spiral. Excel's radar graph plots points increasing from 0.1 to 10.0 with increments of 0.1 and laid out in 10 columns of 10 rows of the spreadsheet. Once you have printed the spiral you can make holes with a drawing pin which will later form the centre pin. The
protractor scale can be copied from a large 360 protractor in pen although this is laborious.

The disk compass enables you to accurately and quickly draw many fascinating diagrams and constructions and Circle Scribe has included many in their book Fun Art and Geometry. The by product is a deeper understanding of and confidence in the basic concepts and theorems of Geometry like angle, radius, diameter, curvature, centre, symmetry, etc etc.

One activity which straddles the divide between Maths and Art is the Cardioid: this is a fascinating and very ancient drawing whose name is derived from its appearance as 3 dimensional heart shape. It is based on a ring of 36 dots evenly spaced around a circle at 10 degree intervals. Each circle has its centre on one of the dots and its circumference passes through the top dot. The envelope is
also the locus of a point on the circumference of a circle rolling around another of the same radius.

There are several similar diagrams based on the ring of dots and the nephroid shown here is named after its kidney shape. In this diagram a vertical diameter is drawn through the ring of dots. Each circle is again centred on a dot but this time the constraint is that they must touch the diameter. There are many lessons about tangents to be learned from constructing this diagram.

Using the coordinate system it is possible to accurately and quickly construct many fascinating diagrams such as the sine wave, the conic sections (parabola, hyperbola, and ellipse) with reference to their directrix and focus.

Their most recent addition is Sierpinski's triangle - a simple fractal which can be drawn very accurately and very quickly. Visit www.fractalus.com to see some computer generated fractals.

Put the Centre pin near the left of the page, turn it this way up, and press on the centre pin to mark its place:

Dot the paper through holes 10, 20, 30, ... 80 to form the bottom rows of dots. Leave the Disk in place.

Mark the protractor zero, turn through 60 degreees and dot the paper through the same holes again to form the dots along the left side.

Put the centre pin at the end dot. Rotate until the 80 hole is over the top dot and mark the paper through the same holes again to form the dots along the right side.

Draw this triangle byjoining the mid points of the sides of the triangle of dots.

Build the diagram by lining a ruler up with the dots and drawing these triangles. Draw as many lines as you can with the ruler in on place.

Put the centre pin at the corners again and this time use the holes 5, 15, ... 75 to make new rows of dots along the sides.

Draw new triangles by joinging these dots to fill out the diagram more. You could go on forever except for the thickness of the pencil line etc.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.